In standard LXDE mode, there is no wireless connection - when clicking the icon of Internet connection, in wireless section there is a label "device is not ready". When activating pns-tool/xpns-tool via terminal, wifi manager does not show any devices as well.

The hardware is netbook Asus EEE with Windows XP, I boot Porteus from pendrive, ext2 file system, produced under Linux Mint.

nonetwork
... This boot parameter will prevent Porteus from automatically
establishing a network connection on start-up.
Note that this cheatcode makes some modifications to
your system which must be manually restored, so it is
recommended that you only use this cheatcode in "Always
Fresh" mode. If you've ignored this advice and need to
recover your network connection, please run the following:
chmod +x /etc/rc.d/rc.inet1 /etc/rc.d/rc.networkmanager
sed -i 's/Hidden=true//g' /etc/xdg/autostart/nm-applet.desktop

This means in the other mode you have made a modification that prevents it from working as it should.

In the second one terminal displayed "no such directory or folder" message...

Please be more specific. What is 'the second one'? If these files or folders don't exist then something is wrong with your installation.

Finally, please boot into the non working mode and use the Porteus Settings Centre to generate a system information dump. Save it somewhere, boot into fresh mode and upload it to pastebin and post the link here.

How do i become super user?
Wear your underpants on the outside and put on a cape.

This means in the other mode you have made a modification that prevents it from working as it should.

I didn't do any changes - it was first start.

In the second one terminal displayed "no such directory or folder" message...
Please be more specific. What is 'the second one'? If these files or folders don't exist then something is wrong with your installation.

Finally, please boot into the non working mode and use the Porteus Settings Centre to generate a system information dump. Save it somewhere, boot into fresh mode and upload it to pastebin and post the link here.